A simplex contained in a sphere (Q1021287)

The author concerns himself with a conjecture (in four different guises) which is related to the generalized Euler's inequality \(R \geq n r\) between circumradius \(R\) and inradius \(r\) of an arbitrary simplex in \(\mathbb E^n.\) Let \(\Delta\) denote an \(n\)-simplex in \(\mathbb E^n\) with the set of vertices \(V = \{v_1, \dots, v_n \}\) and the set of facets \(F = \{f_1,\dots, f_n \}.\) The inradius \(r(\Delta)\) is the maximum of the radii of balls contained within \(\Delta\). A sphere of such maximum radius meets each facet of \(\Delta\) in a single point. On the other hand the circumradius \(R(\Delta)\) is defined here as the minimum of the radii of balls containing \(\Delta.\) The boundary of this minimal ball is a sphere with center \(O\) (circumcenter) but it is not necessarily the sphere passing through the vertices of \(\Delta\), whose radius will be denoted by \(R^{\prime}(\Delta)\) (e.g. for an obtuse triangle in \(\mathbb E^2\)). Let \(\bar\Delta\) be the simplex whose vertices are the feet of the perpendiculars from \(O\) to the facets of \(\Delta.\) Moreover, if \(v \in V\) is a vertex and \(x \in \Delta\) any point in \(\Delta\) denote by \(\Delta_v^x\) the simplex with vertex set \((V-\{v\})\cup \{x \}.\) The author formulates the following four statements (conjectures): (1)\, \(R(\Delta) \geq 2 R(\bar\Delta).\; \) (2)\,If \(\Delta\) is contained in a closed \(n\)-ball \(B,\) then for any point \(x \in \Delta\) there is a facet \(f \in F\) such that \(r_f(x) \in B, \) where \(r_f\) denotes the reflection in a facet \(f.\)\; (3)\, With the same assumption as above, there exists a point \(x \in \Delta\) such that \(r_f(x) \in B\) for all \(f \in F.\)\; (4)\, If \(x \in \Delta,\) then \(R^{\prime}(\Delta ) \leq \max_{x\in V} R^{\prime}(\Delta_v^x).\) The main result of the paper is the proof that these four statements are equivalent, i.e. if any one of the four conjectures is true so are the remaining three. The author proves, moreover, that his conjectures are true for triangles in \(\mathbb E^2\) but the proof (or a counterexample) of any of these conjectures remains elusive in higher dimensions.